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Some Variations on the Theme of Polyominoes


Overview

This topic looks at patterns that illustrate how the squares of polyominoes are connected to each other.


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Below are some examples of Polyominoes of Order 2, 4, 5 and 9.

In the following figures, the darker squares illustrate some different ways that a square in a Polyomino could be connected to the squares that are adjacent to it.

The dark square in the first Polyomino is connected to the square on its right. The dark square in the second Polyomino is connected to squares on the left, right and below. The dark square in the third Polyomino is connected to the square above and the one to the left. The dark square in the last Polyomino in only connected to the square below.


Illustrating Polyomino Connectivity

The following figure shows the same Polyominoes but this time each Polyomino includes 'connection' information - a line joins the mid-point of two adjacent squares if the squares are connected.

Adjacent squares share a common edge but may or may not be connected along that edge. In this case, a collection of adjacent squares is a Polyomino if there is a continuous series of connections joining every pair of squares. By this definition, the figure below on the left represents two Polyominoes and the one on the right represents one.

Connected in this sense could be viewed as physically joined - if you think of the Polyomino constructed out of thin squares of material, with adjacent squares joined along their edges then you have a Polyomino if you can pick up any square and all of the rest of the squares get pulled up too.

The polyominoes below have the same shape but different connectivity patterns.

There are 16 different ways a square can be connected to other squares in a Polyomino (counting rotations as distinct) and below are the connectivity patterns that a square would have for each of these 16 ways. The first square is not connected to any other square. The pattern in the second square shows that it is connected to the square above it... and so on. The last square's pattern shows that it is connected to all four adjacent squares.

These are the order 1, 2 and 3 Polyominoes shown with connectivity information. Rotations and reflections are not shown.

The order 4 Polyominoes... note that the first two Polyominoes have the same shape but different connectivity patterns - because there are two different ways that that particular configuration of four squares can be connected. Again, the rotations and reflections of each polyomino shape are not shown.

The order 5 Polyominoes... all of the ones in the last row have the same shape but different connectivity patterns.

The order 6 Polyominoes.

These order 6 Polyominoes each have more than one connectivity pattern:

These are all of the connectivity patterns for the order 6 Polyomino that has a 2x3 rectangular shape:

Here are some symmetric Polyominoes of order 15 that include one and only one of each of the 15 possible connectivity patterns... are there others?

Here are some non-symmetric ones.

Another Way of Illustrating Polyomino Connectivity

In this section, the connectivity of the Polyominoes is illustrated by by placing equal triangles on each side of a common connecting edge.

First, the original Polyominoes...

The following figure shows the same Polyominoes but this time each Polyomino includes 'connection' information - a pair of equal triangles placed on each side of a common connecting edge.

There are 16 different ways a square can be connected to other squares in a Polyomino (counting rotations as distinct) and the following figure shows the pattern that a square would have for each of these 16 ways. The first square is not connected to any other square. The pattern in the second square shows that it is connected to the square above it... and so on. The last square's pattern shows that it is connected to all four adjacent squares.

These are the order 1, 2 and 3 Polyominoes shown with connectivity information.

The order 4 Polyominoes... note that the first two Polyominoes have the same shape but different connectivity patterns - because there are two different ways that that particular configuration of four squares can be connected.

The order 5 Polyominoes... again, all of the ones in the last row have the same shape but different connectivity patterns.

The order 6 Polyominoes.

These order 6 Polyominoes each have two or more connectivity patterns:

Here are some symmetric Polyominoes of order 15 that include one and only one of each of the 15 possible connectivity patterns... are there others?

Here are some non-symmetric ones.

Polyomino 'Differentiation' and 'Integration'

One interesting characteristic of this way of illustrating Polyomino connectivity is that the connectivity pattern is itself a Polyomino (or set of Polyominoes). So, using an analogy from calculus, the process of generating the connectivity pattern for a particular polyomino could be called 'differentiation'... Doing the reverse, ie finding the Polyomino(s) that differentiate to a particular Polyomino could be called 'integration'.


The following figure shows an order 5 Polyomino and the results of successive differentiation operations.

The next figure shows an order 4 Polyomino and the two possible order 5 Polyominoes that it integrates to.

Polyominoes that 'Differentiate' to Themselves

Actually only the first order four Polyomino really differentiates to itself... the others differentiate to a mirror image of themselves.

Polyominoes that 'Differentiate' to Themselves in Two Steps

This is one example... are there others?

Polyominoes that grow as they 'Differentiate'

A Polyomino that has a certain minimum shape (eg contains at least a 2 by 6 rectangle) will grow larger in successive differentiation operations. The following figure illustrates this.

The Polyominoes in this series have sizes (ie number of squares)
6, 7, 8, 10, 12, 16, 20, 28...


Some order 9 polyominoes with a square 3x3 shape and their corresponding connectivity patterns:
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